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This paper concerns the unscented Kalman filter (UKF) for the nonlinear dynamic systems. The sampling principle of UKF is firstly addressed, which is based on moment matching method. Then we designed an extended symmetric sampling strategy, given an n-dimensional state, which defines 4n+1 symmetric points that lie on axes to fully represent the mean and covariance of the state. The performance of the two UKFs, namely, the UKF and the extended symmetric UKF (EUKF), is compared by using the mean of the root of mean square error. The simulation results showed that EUKF outperforms the UKF in the presence of strong noise and the scalar k is a key factor involved in both UKFs.